(a) What does that expression tend toward as r becomes infinite ?

One ie. âˆ†Ï„/âˆ†t = 1

Quote:

(b) What does that expression tend toward as r goes toward R_s, while always r > R_s ?

Zero ie. âˆ†Ï„/âˆ†t -> 0

Quote:

(c) For (a) and (b) what does that mean about the passage of time for LowGuy as viewed by ( well out into Euclidean space ) HighGal ?

When LowGuy is up near HighGal their clock rates agree.
As HighGal views LowGuy's clock when he approaches the event horizon his appears to slow down ( more on this later ).

Quote:

(d) BONUS POINTS : What if r = R_s ?

Zero ie. âˆ†Ï„/âˆ†t = 0

LowGuy's clock stops.

Quote:

(e) DOUBLE BONUS POINTS : What if r < R_s ? { HINT : we are using the squares of time intervals both sides, so be careful .... :-) }

We used the square root to get the time rate ratio, so if we stick with real numbers then there is no solution via Schwarzchild's approach for the time rate closer in than the hozizon as seen from the outside. That is you can't take the square root of a negative number and get a real number.

Quote:

(f) MEGA BONUS POINTS : Does this work for the Sun, or me, or you too ? :-))

If we were all black holes ( all the relevant mass within the respective radii ) then yes ! :-))

Evidently then this R_s as emitted from Karl''s solution comes to physical significance when measuring relative time rates of the clocks as defined. This R_s as already indicated is where the event horizon is. As discussed this is spherically symmetric and thus defines a shell shape. We cannot get any info from within.

But Then Again ...

There is another way of solving the GR equations which still yields all of the above. This was discovered by Kruskal et al. But it doesn't use anything as simple as Karl's concentric spheres, so I won't go anywhere near describing it formally as I have no idea whatsoever. It examines the quite reasonable question : what would be the experience of an inbound traveller as they cross the horizon at R_s ?

The amazing answer is that generally nothing especial happens in the immediate zone ( in time and space ) for the traveller. For rather small holes the gravity gradient ( strength of gravity change per distance ) can be quite savage and so the feet of a traveller will accelerate rather differently than their head ( if they are descending feet first ). Not a good survival prospect. You get ripped up. The differential at the horizon is more gentler with a larger black hole, and the general case is the spacetime curvature at R_s goes like 1/M^2, or 1/R_s^2 if you like. For the falling traveller there is no physical event/phenomena that marks the crossing at R_s, as compared to happenings just before or just after.

For instance it seems the centre of our galaxy, and many others, has a central black hole containing the mass of many millions of suns. So the R_s is in the range of millions of kilometers, and the rate at which field strength changes is much gentler indeed.

The Kruskal solution gives 'sensible' answers right up until radius of zero. Several points :

- the 'radius' here is not really a coordinate type you could qualitatively translate properly to outside the hole. Without getting into nuances we will just call radius = 0 the centre of the hole.

- this is where every prediction gives an infinite answer, even for a traveller who is measuring something prior to hitting it.

- inside R_s the radius must always decrease with proper time ie. if we threw LowGuy in he would experience a finite period of time before hitting the centre. Outside of R_s we have the possibility of staying constant or increasing radius at least.

- the technical term for this is singularity, a word that has more a mathematical pedigree than physical. It really means 'no solution' for our questions, with the physical behaviour that everything within the horizon will end up there. All stacked up at one point. At least this is the pure GR answer without accounting for QM etc.

BTW : I've decided not to use LowGuy for this bit. It would be murder and in any event ( pardon the pun ) we'd never get a report. So let's drop only his clock ( LowGuy's Clock or LGC ) into the hole instead, suitably setup to periodically ( from it's rest frame ) emit photons. These photons we will allow to spray in all directions from the clock. Let us discuss what happens to them. The paths these photons go through are called 'light lines' or 'world lines of photons/light' etc. They 'map' the distortion of spacetime. Because all particles with non-zero rest mass ( you & me folks ) go slower than light then such paths bound material movements.

Questions 5

Put a source of photons exactly at the horizon. Let it spray photons in every direction. What will happen to photons that :

(a) Go radially outwards ?

(b) Go along a tangent to the radial direction ?

(c) Go inwards ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

It's worth outlining this some more. Most of you are probably sitting in a room when reading this. Let's pretend for a moment that over the extents of the room there is a discernible difference in the strength of gravity ie. you could feel it yourself without a gadget to help you measure. So you could walk around the room and feel, say, that close to the Ronald McDonald portrait you felt a bit lighter whereas nearby the filing cabinet you feel a bit heavier. What could you make of this as in the ordinary course of events this would be silly, right ? Now if you knew that you weren't rotating ( the cues are easy if you look outside ) then you might hypothecate that something really, really heavy was under the filing cabinet. Such a big localised mass would have other effects. Spill some water on the lovely marble floor and it starts to creep away from Ronnie and towards your tax returns. :-)

This is an example of tidal behaviour and is the key feature of gravity. It means that the strength of gravity varies with position. Of course it does, you say, what force does not ? Which is quite right as all force laws have some distance dependency, even the nuclear ones. But gravity is universal because everything is either mass/energy, but not all have electric charge and/or the various nuclear force attributes. Plus gravity is long range and so it acts to compress all bodies in that fashion. Gravity is always attractive. There is deep importance in the fact that we measure gravitational field strength ( force per unit mass ) in units of acceleration. So a neutron in a neutron star is attracted by all other neutrons in the star but is only repulsed by relatively few neighbours. Such local repulsion is not altered by adding other neutrons to the pile, except perhaps indirectly by pulling them all closer together. Gravity thus accumulates and will always eventually beat the character of locally ranged forces in this universe.

So what keeps us from having a liquid/gaseous structure is inter-molecular forces. These are essentially electromagnetic + quantum mechanical rules dictating a 'sweet spot' or optimal distances b/w according to the molecules in question. These are longer term arrangements, as opposed to brief bounces in fluids, where specific individual entities spend time near other specific individual identities. This is structure of course that the solid phase represents. This is in practice rather complicated and certainly a long way from pure Coulomb interaction : which is still true as a one-on-one law, but in the presence of many objects yields curious distance dependencies. Nuclei contain the positive charges and are surrounded by a fog of whizzing electrons having negative charge. The general character is :

- far away no force. At distance the positive/negative charges appear as so close together that the nett is effectively no charge and no ( static ) EM effect.

- closer in there are polar patterns eg. dipole which means the charges re-arrange to give corresponding opposite charge concentrations as adjacent and hence nett attraction.

- really close in. The electron clouds abut to closer proximity than the nuclei and hence nett repulsion.

- in between is the best of the Three Bears and that gives the wonderful range of persistent arrangements that charactises many of the constant-ish things in life. Including life itself.

Back at the office the various EM bonds b/w your body parts are challenged in the presence of gravity to the extent of keeping you upright and intact in a uniform gravitational field. That works up to a point and depends upon what we mean by uniform. This is in truth an approximation so normally we could say there is not much difference b/w the heels and the head.

Down at the Schwarzchild radius ( but also maybe other places too ) if the different parts of a body are acted upon with different accelerations then those parts will want to change their relative separations. If not otherwise restrained .....

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Ok. You certainly don't see any of them distantly ie. from HighGal's position :

Quote:

(a) Go radially outwards ?

These you will get to see in an infinite amount of time ! Or if you like the photons have an infinite red shift - lowering of frequency to zero. Another idea is that they 'hover' at the horizon, being frozen in time they don't move.

Quote:

(b) Go along a tangent to the radial direction ?

These will scoot along around the horizon, remaining at the Schwarzschild radius and circulating forever.

Quote:

(c) Go inwards ?

They spiral inwards to the central singularity.

So yes : these are all equally unmeasurable/unobservable to HighGal.

If the photons were emitted just outside of the horizon, some would be potentially eventually seen by HighGal in a finite time ( well, how long will the Universe persist for ? ).

Questions 6

How/why would a photon emitted just outside the horizon get to an observer far away ?

{ HINT : direction }

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

If the photon was emitted radially then it would continue in the same direction with a much reduced frequency. In any other 'upwards' direction it would spiral around the singularity until it was far enough away to escape albeit with a much reduced frequency. What this would look like to HighGal?

If the photon was emitted radially then it would continue in the same direction with a much reduced frequency. In any other 'upwards' direction it would spiral around the singularity until it was far enough away to escape albeit with a much reduced frequency. What this would look like to HighGal?

Correct. If emitted in some sense away from the hole it would eventually emerge to be viewed a long time later and with depleted frequency/energy.

When I get my post flu ducks in a row. I'll recap and then move on to the rotation aspect.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Recap : A static black hole. It has a spherical horizon that is evident when viewed from far away. Objects falling to that horizon appear to have slower rate of time passage ( proper time ), in a limiting way such that they never appear to finally reach the horizon and their emitted photons lose energy and appear red-shifted ( to lower frequency ). This Schwarzschild solution gives no answer for what happens inside the hole, or if you like nothing can be said as the light can't get out.

This red shift increases the closer one is to the horizon and you could call this effect the curvature of time. Mathematically that is evident as the ratio of the interval of one time tick on a descending clock relative to one time tick on a distant clock.

Radial distances are also affected as the horizon is approached. Concentric shells are closer than they ought be ( with respect to distant space ) according to their circumferences/areas. This proper length increases without limit the closer to the horizon. This effect could be called the curvature of space, as is evident in the ratio of radial lengths b/w a descending and a distant observer.

What Generates Gravity Anyway?

In Newton's day this was easy. It's mass. It's the stuff that the universe is made of. The idea of mass, inertia and even momentum were assumed as axioms without much closer inspection. Anyone reading his Principia had a pretty good feeling or instinct for that. An object's mass did not change depending upon the viewpoint, and specifically not upon velocity.

Cue Einstein. When his GR equations are viewed, well, in generality ( not in any specific applied circumstance or solution ) one can tease out types of sources of gravitation. These are most apparent in the low field situation. In detail this is a messy mathematical affair, but I hope we can take a hint from analysis thus far. By example here is the full metric for the Schwarzschild analysis ( don't panic ) :

[pre][âˆ†s]^2 = [1 - (R_s/r)] * (âˆ†t)^2 { curvature of time }

+ (âˆ†r)^2 / [1 - (R_s/r)] { curvature of space in the radial direction }

+ r^2 [(âˆ†Î¸)^2 + (sinÎ¸)^2 * (âˆ†Ï†)^2] { a Euclidean spherical shell }
[/pre]
... where I have introduced âˆ†s as the spacetime interval, which represents a span in space or time or a mixture, depending upon choices of âˆ†Î¸, âˆ†Ï† and âˆ†r. So if I set âˆ†Î¸, âˆ†Ï† and âˆ†r all to zero ( meaning that I stay in the one position ) then only the time term remains on the right hand side of the equation and so âˆ†s represents time too. Anyway for this scenario there is a part that describes time curvature and a bit that describes spatial curvature. This is true for other solutions in other situations.

Time curvature is caused by what is deemed 'active gravitational mass'. This has several components including all the ( rest ) mass, the mass equivalent of any 'pure' energy like photons, plus a pressure term. So the Newtonian part is in that plus extras. We have to do mass equivalent because of special relativity deems that has inertial mass, and the equivalence principle asserts/confirms that inertial mass is gravitational too. Beware the pressure bit. This is like the internal pressure of a gas ( in fact it may be exactly that ), the banging about of the molecules within some volume. It isn't the bulk movement of the gas in some nett direction. So if I stand outdoors on a windless day, then I suffer only the uniform pressure of the atmosphere upon my body. If the wind kicks up then there is a 'ram' component, velocity dependent, as well as the static bit. Ride a motor bike, or stick you head/hand out the car window and that's the ram bit also. We're not including that dynamic pressure here.

The pressure term contributing to active gravitational mass enters with a plus sign, a multiplier of three, but is divided by c^2 and thus is normally not noticeable. Indeed it wasn't noticed until we discovered objects like neutron stars. Up to some point they don't collapse further because of internal pressure ( nuclear force repelling whatever the central components of such objects truly are ). Now the Catch-22 : the more pressure, the more gravity from said pressure and that pulls stuff inwards increasing the pressure yet again. The neutron star can never win here.

{ I say 'beware pressure' as this can be set as negative ie. inward pulling rather than outward pushing. This leads to that weird cosmic stuff with expansion of the universe assisted by what is effectively a repulsive component to gravity. This is not intuitive but is predicted by the equations. Let's not go there. }

{ ASIDE : due to a formal analogy with electromagnetism - the Coulomb field around a static electric charge - this time curvature component is also called the 'gravitoelectric' part of the gravitational field. }

Spatial curvature is caused by what is deemed as 'active curvature mass'. This also is the sum of mass/energy and a pressure term. The pressure term enters with a minus sign, a multiplier of one (1), though also is suppressed by the factor of c^2. Again the pressure part was not evident before Einstein's theory.

Thus far we have avoided nett momentum ( the ram bit ). How does this work ? Imagine a large body rushing by you in empty space. It will attract you by it's Newtonian gravity in a direction toward that body ( even including a time delay ). But Newton only admits a rest mass. For Isaac there was no other type. This moving body attracting you has more mass than rest mass, because it is not at rest ! If you like the body's kinetic energy, the energy of relative motion, has it's own mass equivalent. Which must gravitate, yes ?

{ ASIDE : Yep, you guessed it! This is the 'gravitomagnetism' part, so named due to analogy with electromagnetism. There is lots of hand waving, both left and right hand rules apply in determining the direction of effects, but it all comes out in the wash. }

These are all brutal conclusions.

I mean brutal because if you swallow a bit of Einstein's GR then you must swallow the whole. You can't take a bit that you like and ignore the rest. You have to follow the logic outward from the base precepts. It's a package. The glue in the package is called the principle of general covariance. It is a principle that has physical meaning and mathematical expression. Different observers, because they are viewing a common reality, have to ( eventually ) agree upon outcomes of processes. One observer cannot say the star didn't collapse, when another did. It may take time to reconcile the versions as light speed is finite. If observers can never communicate ( LowGuy disappears past the horizon ) then they are at liberty to disagree. So the three sources of gravity thus far mentioned 'total up' to the same outcomes regardless of the viewer, even though different viewers may attribute different values to the components. This general covariance is a solid principle. In a sense it is a meta-principle, a principle about principles*. You could label it as an item of philosophy, one that prevents contradictions.

{ Indeed not allowing contradictory versions of the one universe was coined as the 'cosmic censorship principle' by Penrose et al. It is a ( post-hoc ) idea that forces horizons to hide black hole interiors. }

One last gulp. In fact it is the biggest chunk in some ways, an abstraction that takes some willing to deal with. The gravitational field itself has a gravitational effect. Arrggghh, you say ! In Newton's day and onwards the field was somewhat of an artifice. A useful premise for calculation to distinguish the sources of gravitation from the effect on another mass at some chosen point. The field 'transmitted' the force, disposed of the action-at-a-distance concept etc. It is best/easiest to accept the field as real, a beast in and of itself. As such it must contain energy. Do you remember potential energy from Newtonian mechanics ? I climb a ladder while expending energy to do so, working against the force of gravity that ever pulls me to the bottom of the ladder. At the top of that ladder, and at rest, I have energy in the bank. You can't see it, but you will if I jump off. Then I convert potential to kinetic on the way down. At the top before I stepped off the energy was not mine alone though. It can't be. One may have ladders on many planets with me up the top & the same thing will happen but to different degrees. For a given ladder height I won't be going as fast when I reach the surface of Mars compared to Earth, and less again for the Moon. Right down to zero for a me plus ladder combination in free space. I can flip the context here and be totally Mike-centric and insist that the Earth came towards me when I ceased contact with the ladder rungs. Mutatis mutandis for Mars & Moon. So the potential energy doesn't belong to any celestial body either. It is a function of the interaction which is another way of saying that the field has said energy. Yup. Cue E = m*c^2, or m = E / c^2 and that has a mass equivalent .....

.... which means that when you 'solve' for the covariant parts, as above, then you are not finished. You have to plug in this extra field part and go around again. Maybe again. And again. This is the wicked non-linear bit. You would hope that the corrections become less and less with iteration. But this is also why, when two black holes collide, some energy from that field comes our way and slightly wiggles our instruments ! :-)

{ That's why most GR solutions are not analytic. Supercomputers are your best mates here ie. numerical relativity. }

Ok. Nuff for now. Next up we will look at the rotating/gravitomagnetic aspect.

Cheers, Mike.

* Without getting too fancy, you could view the principle of general covariance as a heuristic also. Much has been made of it's exact status back in 1915, as now. IMHO it's best to relax about which precise category of thought we should place it in. I'd call it a great inspiration on the day and leave it at that. :-)

( edit ) Any questions at all ?

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

OK, now for the topic of rotation or frame dragging. In detail this is even more horrible maths. The metric expression ( in a spherical polar system for instance ) has terms like âˆ†r * âˆ†Î¸. In effect this means that spacetime traverses of constant_something have an offset in radius combined with an offset in longitudinal angle, say. You could physically interpret this by saying that things vary in the known way with radius ( as per the static case ) but only if you rotate a bit every time you move in/out a bit in the radial direction. Mentally this gives the picture of a ( previously ) radially outward line now swirling around in a cyclonic pattern.

In fact the cyclone idea is not a bad one. I am reminded of one of the Pirates Of The Caribbean movies that had a 'maelstrom', that being like a funnel of water going down a plughole in the ocean. You could also think of the central funnel of a twister/hurricane. The key feature here is that one cannot stay at some fixed angle in longitude : you have to rotate with the general flow no matter what you do.

How so for light then ? A photon has to follow likewise. For to go against the flow would require faster than light travel and not even photons can do that. We spoke before of photons at the horizon 'hovering' at that radius ( as analysed distantly by a limiting argument ). Thus photons will swirl in the direction of a black hole's rotation.

Are we now in a position to explain at least some of the features in the original frame ? I think so. For starters : who wants to punt in idea(s) as to why there is a central volume of distortion evident :

.... submit answers as usual on the back of $100 note. :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Back to task. The central area - of whatever arrangement - is going to gravitate regardless. That is, from a distance massive bodies and light will experience attraction in a focusing manner regardless of whether a black hole has formed, how many there are, and what they are up to amongst themselves.

The next aspect :

... is also straightforward. The larger mass hole has the larger apparent diameter. I say 'apparent' given the various qualifying comments as discussed. Now why do the merging holes :

... seem to avoid each other in between ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Ok : [pre]1 -

)

Ok :

[pre]1 - (R_s/r)[/pre]

One ie. âˆ†Ï„/âˆ†t = 1

Zero ie. âˆ†Ï„/âˆ†t -> 0

When LowGuy is up near HighGal their clock rates agree.

As HighGal views LowGuy's clock when he approaches the event horizon his appears to slow down ( more on this later ).

Zero ie. âˆ†Ï„/âˆ†t = 0

LowGuy's clock stops.

We used the square root to get the time rate ratio, so if we stick with real numbers then there is no solution via Schwarzchild's approach for the time rate closer in than the hozizon as seen from the outside. That is you can't take the square root of a negative number and get a real number.

If we were all black holes ( all the relevant mass within the respective radii ) then yes ! :-))

Evidently then this R_s as emitted from Karl''s solution comes to physical significance when measuring relative time rates of the clocks as defined. This R_s as already indicated is where the event horizon is. As discussed this is spherically symmetric and thus defines a shell shape. We cannot get any info from within.

But Then Again ...

There is another way of solving the GR equations which still yields all of the above. This was discovered by Kruskal et al. But it doesn't use anything as simple as Karl's concentric spheres, so I won't go anywhere near describing it formally as I have no idea whatsoever. It examines the quite reasonable question : what would be the experience of an inbound traveller as they cross the horizon at R_s ?

The amazing answer is that generally nothing especial happens in the immediate zone ( in time and space ) for the traveller. For rather small holes the gravity gradient ( strength of gravity change per distance ) can be quite savage and so the feet of a traveller will accelerate rather differently than their head ( if they are descending feet first ). Not a good survival prospect. You get ripped up. The differential at the horizon is more gentler with a larger black hole, and the general case is the spacetime curvature at R_s goes like 1/M^2, or 1/R_s^2 if you like. For the falling traveller there is no physical event/phenomena that marks the crossing at R_s, as compared to happenings just before or just after.

For instance it seems the centre of our galaxy, and many others, has a central black hole containing the mass of many millions of suns. So the R_s is in the range of millions of kilometers, and the rate at which field strength changes is much gentler indeed.

The Kruskal solution gives 'sensible' answers right up until radius of zero. Several points :

- the 'radius' here is not really a coordinate type you could qualitatively translate properly to outside the hole. Without getting into nuances we will just call radius = 0 the centre of the hole.

- this is where every prediction gives an infinite answer, even for a traveller who is measuring something prior to hitting it.

- inside R_s the radius must always decrease with proper time ie. if we threw LowGuy in he would experience a finite period of time before hitting the centre. Outside of R_s we have the possibility of staying constant or increasing radius at least.

- the technical term for this is singularity, a word that has more a mathematical pedigree than physical. It really means 'no solution' for our questions, with the physical behaviour that everything within the horizon will end up there. All stacked up at one point. At least this is the pure GR answer without accounting for QM etc.

BTW : I've decided not to use LowGuy for this bit. It would be murder and in any event ( pardon the pun ) we'd never get a report. So let's drop only his clock ( LowGuy's Clock or LGC ) into the hole instead, suitably setup to periodically ( from it's rest frame ) emit photons. These photons we will allow to spray in all directions from the clock. Let us discuss what happens to them. The paths these photons go through are called 'light lines' or 'world lines of photons/light' etc. They 'map' the distortion of spacetime. Because all particles with non-zero rest mass ( you & me folks ) go slower than light then such paths bound material movements.

Questions 5

Put a source of photons exactly at the horizon. Let it spray photons in every direction. What will happen to photons that :

(a) Go radially outwards ?

(b) Go along a tangent to the radial direction ?

(c) Go inwards ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: You get ripped

)

It's worth outlining this some more. Most of you are probably sitting in a room when reading this. Let's pretend for a moment that over the extents of the room there is a discernible difference in the strength of gravity ie. you could feel it yourself without a gadget to help you measure. So you could walk around the room and feel, say, that close to the Ronald McDonald portrait you felt a bit lighter whereas nearby the filing cabinet you feel a bit heavier. What could you make of this as in the ordinary course of events this would be silly, right ? Now if you knew that you weren't rotating ( the cues are easy if you look outside ) then you might hypothecate that something really, really heavy was under the filing cabinet. Such a big localised mass would have other effects. Spill some water on the lovely marble floor and it starts to creep away from Ronnie and towards your tax returns. :-)

This is an example of tidal behaviour and is the key feature of gravity. It means that the strength of gravity varies with position. Of course it does, you say, what force does not ? Which is quite right as all force laws have some distance dependency, even the nuclear ones. But gravity is universal because everything is either mass/energy, but not all have electric charge and/or the various nuclear force attributes. Plus gravity is long range and so it acts to compress all bodies in that fashion. Gravity is always attractive. There is deep importance in the fact that we measure gravitational field strength ( force per unit mass ) in units of acceleration. So a neutron in a neutron star is attracted by all other neutrons in the star but is only repulsed by relatively few neighbours. Such local repulsion is not altered by adding other neutrons to the pile, except perhaps indirectly by pulling them all closer together. Gravity thus accumulates and will always eventually beat the character of locally ranged forces in this universe.

So what keeps us from having a liquid/gaseous structure is inter-molecular forces. These are essentially electromagnetic + quantum mechanical rules dictating a 'sweet spot' or optimal distances b/w according to the molecules in question. These are longer term arrangements, as opposed to brief bounces in fluids, where specific individual entities spend time near other specific individual identities. This is structure of course that the solid phase represents. This is in practice rather complicated and certainly a long way from pure Coulomb interaction : which is still true as a one-on-one law, but in the presence of many objects yields curious distance dependencies. Nuclei contain the positive charges and are surrounded by a fog of whizzing electrons having negative charge. The general character is :

- far away no force. At distance the positive/negative charges appear as so close together that the nett is effectively no charge and no ( static ) EM effect.

- closer in there are polar patterns eg. dipole which means the charges re-arrange to give corresponding opposite charge concentrations as adjacent and hence nett attraction.

- really close in. The electron clouds abut to closer proximity than the nuclei and hence nett repulsion.

- in between is the best of the Three Bears and that gives the wonderful range of persistent arrangements that charactises many of the constant-ish things in life. Including life itself.

Back at the office the various EM bonds b/w your body parts are challenged in the presence of gravity to the extent of keeping you upright and intact in a uniform gravitational field. That works up to a point and depends upon what we mean by uniform. This is in truth an approximation so normally we could say there is not much difference b/w the heels and the head.

Down at the Schwarzchild radius ( but also maybe other places too ) if the different parts of a body are acted upon with different accelerations then those parts will want to change their relative separations. If not otherwise restrained .....

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: Questions 5 Put a

)

Sorry for not replying earlier. I was drunk.

The answer. Nobody knows because it is unobservable.

Richard

## Ok. You certainly don't see

)

Ok. You certainly don't see any of them distantly ie. from HighGal's position :

These you will get to see in an infinite amount of time ! Or if you like the photons have an infinite red shift - lowering of frequency to zero. Another idea is that they 'hover' at the horizon, being frozen in time they don't move.

These will scoot along around the horizon, remaining at the Schwarzschild radius and circulating forever.

They spiral inwards to the central singularity.

So yes : these are all equally unmeasurable/unobservable to HighGal.

If the photons were emitted just outside of the horizon, some would be potentially eventually seen by HighGal in a finite time ( well, how long will the Universe persist for ? ).

Questions 6

How/why would a photon emitted just outside the horizon get to an observer far away ?

{ HINT : direction }

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## If the photon was emitted

)

If the photon was emitted radially then it would continue in the same direction with a much reduced frequency. In any other 'upwards' direction it would spiral around the singularity until it was far enough away to escape albeit with a much reduced frequency. What this would look like to HighGal?

Richard

## RE: If the photon was

)

Correct. If emitted in some sense away from the hole it would eventually emerge to be viewed a long time later and with depleted frequency/energy.

When I get my post flu ducks in a row. I'll recap and then move on to the rotation aspect.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Recap : A static black hole.

)

Recap : A static black hole. It has a spherical horizon that is evident when viewed from far away. Objects falling to that horizon appear to have slower rate of time passage ( proper time ), in a limiting way such that they never appear to finally reach the horizon and their emitted photons lose energy and appear red-shifted ( to lower frequency ). This Schwarzschild solution gives no answer for what happens inside the hole, or if you like nothing can be said as the light can't get out.

This red shift increases the closer one is to the horizon and you could call this effect the curvature of time. Mathematically that is evident as the ratio of the interval of one time tick on a descending clock relative to one time tick on a distant clock.

Radial distances are also affected as the horizon is approached. Concentric shells are closer than they ought be ( with respect to distant space ) according to their circumferences/areas. This proper length increases without limit the closer to the horizon. This effect could be called the curvature of space, as is evident in the ratio of radial lengths b/w a descending and a distant observer.

What Generates Gravity Anyway?

In Newton's day this was easy. It's mass. It's the stuff that the universe is made of. The idea of mass, inertia and even momentum were assumed as axioms without much closer inspection. Anyone reading his Principia had a pretty good feeling or instinct for that. An object's mass did not change depending upon the viewpoint, and specifically not upon velocity.

Cue Einstein. When his GR equations are viewed, well, in generality ( not in any specific applied circumstance or solution ) one can tease out types of sources of gravitation. These are most apparent in the low field situation. In detail this is a messy mathematical affair, but I hope we can take a hint from analysis thus far. By example here is the full metric for the Schwarzschild analysis ( don't panic ) :

[pre][âˆ†s]^2 = [1 - (R_s/r)] * (âˆ†t)^2 { curvature of time }

+ (âˆ†r)^2 / [1 - (R_s/r)] { curvature of space in the radial direction }

+ r^2 [(âˆ†Î¸)^2 + (sinÎ¸)^2 * (âˆ†Ï†)^2] { a Euclidean spherical shell }

[/pre]

... where I have introduced âˆ†s as the spacetime interval, which represents a span in space or time or a mixture, depending upon choices of âˆ†Î¸, âˆ†Ï† and âˆ†r. So if I set âˆ†Î¸, âˆ†Ï† and âˆ†r all to zero ( meaning that I stay in the one position ) then only the time term remains on the right hand side of the equation and so âˆ†s represents time too. Anyway for this scenario there is a part that describes time curvature and a bit that describes spatial curvature. This is true for other solutions in other situations.

Time curvature is caused by what is deemed 'active gravitational mass'. This has several components including all the ( rest ) mass, the mass equivalent of any 'pure' energy like photons, plus a pressure term. So the Newtonian part is in that plus extras. We have to do mass equivalent because of special relativity deems that has inertial mass, and the equivalence principle asserts/confirms that inertial mass is gravitational too. Beware the pressure bit. This is like the internal pressure of a gas ( in fact it may be exactly that ), the banging about of the molecules within some volume. It isn't the bulk movement of the gas in some nett direction. So if I stand outdoors on a windless day, then I suffer only the uniform pressure of the atmosphere upon my body. If the wind kicks up then there is a 'ram' component, velocity dependent, as well as the static bit. Ride a motor bike, or stick you head/hand out the car window and that's the ram bit also. We're not including that dynamic pressure here.

The pressure term contributing to active gravitational mass enters with a plus sign, a multiplier of three, but is divided by c^2 and thus is normally not noticeable. Indeed it wasn't noticed until we discovered objects like neutron stars. Up to some point they don't collapse further because of internal pressure ( nuclear force repelling whatever the central components of such objects truly are ). Now the Catch-22 : the more pressure, the more gravity from said pressure and that pulls stuff inwards increasing the pressure yet again. The neutron star can never win here.

{ I say 'beware pressure' as this can be set as negative ie. inward pulling rather than outward pushing. This leads to that weird cosmic stuff with expansion of the universe assisted by what is effectively a repulsive component to gravity. This is not intuitive but is predicted by the equations. Let's not go there. }

{ ASIDE : due to a formal analogy with electromagnetism - the Coulomb field around a static electric charge - this time curvature component is also called the 'gravitoelectric' part of the gravitational field. }

Spatial curvature is caused by what is deemed as 'active curvature mass'. This also is the sum of mass/energy and a pressure term. The pressure term enters with a minus sign, a multiplier of one (1), though also is suppressed by the factor of c^2. Again the pressure part was not evident before Einstein's theory.

Thus far we have avoided nett momentum ( the ram bit ). How does this work ? Imagine a large body rushing by you in empty space. It will attract you by it's Newtonian gravity in a direction toward that body ( even including a time delay ). But Newton only admits a rest mass. For Isaac there was no other type. This moving body attracting you has more mass than rest mass, because it is not at rest ! If you like the body's kinetic energy, the energy of relative motion, has it's own mass equivalent. Which must gravitate, yes ?

{ ASIDE : Yep, you guessed it! This is the 'gravitomagnetism' part, so named due to analogy with electromagnetism. There is lots of hand waving, both left and right hand rules apply in determining the direction of effects, but it all comes out in the wash. }

These are all brutal conclusions.

I mean brutal because if you swallow a bit of Einstein's GR then you must swallow the whole. You can't take a bit that you like and ignore the rest. You have to follow the logic outward from the base precepts. It's a package. The glue in the package is called the principle of general covariance. It is a principle that has physical meaning and mathematical expression. Different observers, because they are viewing a common reality, have to ( eventually ) agree upon outcomes of processes. One observer cannot say the star didn't collapse, when another did. It may take time to reconcile the versions as light speed is finite. If observers can never communicate ( LowGuy disappears past the horizon ) then they are at liberty to disagree. So the three sources of gravity thus far mentioned 'total up' to the same outcomes regardless of the viewer, even though different viewers may attribute different values to the components. This general covariance is a solid principle. In a sense it is a meta-principle, a principle about principles*. You could label it as an item of philosophy, one that prevents contradictions.

{ Indeed not allowing contradictory versions of the one universe was coined as the 'cosmic censorship principle' by Penrose et al. It is a ( post-hoc ) idea that forces horizons to hide black hole interiors. }

One last gulp. In fact it is the biggest chunk in some ways, an abstraction that takes some willing to deal with. The gravitational field itself has a gravitational effect. Arrggghh, you say ! In Newton's day and onwards the field was somewhat of an artifice. A useful premise for calculation to distinguish the sources of gravitation from the effect on another mass at some chosen point. The field 'transmitted' the force, disposed of the action-at-a-distance concept etc. It is best/easiest to accept the field as real, a beast in and of itself. As such it must contain energy. Do you remember potential energy from Newtonian mechanics ? I climb a ladder while expending energy to do so, working against the force of gravity that ever pulls me to the bottom of the ladder. At the top of that ladder, and at rest, I have energy in the bank. You can't see it, but you will if I jump off. Then I convert potential to kinetic on the way down. At the top before I stepped off the energy was not mine alone though. It can't be. One may have ladders on many planets with me up the top & the same thing will happen but to different degrees. For a given ladder height I won't be going as fast when I reach the surface of Mars compared to Earth, and less again for the Moon. Right down to zero for a me plus ladder combination in free space. I can flip the context here and be totally Mike-centric and insist that the Earth came towards me when I ceased contact with the ladder rungs. Mutatis mutandis for Mars & Moon. So the potential energy doesn't belong to any celestial body either. It is a function of the interaction which is another way of saying that the field has said energy. Yup. Cue E = m*c^2, or m = E / c^2 and that has a mass equivalent .....

.... which means that when you 'solve' for the covariant parts, as above, then you are not finished. You have to plug in this extra field part and go around again. Maybe again. And again. This is the wicked non-linear bit. You would hope that the corrections become less and less with iteration. But this is also why, when two black holes collide, some energy from that field comes our way and slightly wiggles our instruments ! :-)

{ That's why most GR solutions are not analytic. Supercomputers are your best mates here ie. numerical relativity. }

Ok. Nuff for now. Next up we will look at the rotating/gravitomagnetic aspect.

Cheers, Mike.

* Without getting too fancy, you could view the principle of general covariance as a heuristic also. Much has been made of it's exact status back in 1915, as now. IMHO it's best to relax about which precise category of thought we should place it in. I'd call it a great inspiration on the day and leave it at that. :-)

( edit ) Any questions at all ?

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## OK, now for the topic of

)

OK, now for the topic of rotation or frame dragging. In detail this is even more horrible maths. The metric expression ( in a spherical polar system for instance ) has terms like âˆ†r * âˆ†Î¸. In effect this means that spacetime traverses of constant_something have an offset in radius combined with an offset in longitudinal angle, say. You could physically interpret this by saying that things vary in the known way with radius ( as per the static case ) but only if you rotate a bit every time you move in/out a bit in the radial direction. Mentally this gives the picture of a ( previously ) radially outward line now swirling around in a cyclonic pattern.

In fact the cyclone idea is not a bad one. I am reminded of one of the Pirates Of The Caribbean movies that had a 'maelstrom', that being like a funnel of water going down a plughole in the ocean. You could also think of the central funnel of a twister/hurricane. The key feature here is that one cannot stay at some fixed angle in longitude : you have to rotate with the general flow no matter what you do.

How so for light then ? A photon has to follow likewise. For to go against the flow would require faster than light travel and not even photons can do that. We spoke before of photons at the horizon 'hovering' at that radius ( as analysed distantly by a limiting argument ). Thus photons will swirl in the direction of a black hole's rotation.

Are we now in a position to explain at least some of the features in the original frame ? I think so. For starters : who wants to punt in idea(s) as to why there is a central volume of distortion evident :

.... submit answers as usual on the back of $100 note. :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Back to task. The central

)

Back to task. The central area - of whatever arrangement - is going to gravitate regardless. That is, from a distance massive bodies and light will experience attraction in a focusing manner regardless of whether a black hole has formed, how many there are, and what they are up to amongst themselves.

The next aspect :

... is also straightforward. The larger mass hole has the larger apparent diameter. I say 'apparent' given the various qualifying comments as discussed. Now why do the merging holes :

... seem to avoid each other in between ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Do they? This is

)

Do they? This is counter-intuitive.

Richard